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A070825 One half of product of first n+1 Lucas numbers A000032.

%I A070825 #34 Jul 07 2025 15:40:04
%S A070825 1,1,3,12,84,924,16632,482328,22669416,1722875616,211913700768,
%T A070825 42170826452832,13579006117811904,7074662187380001984,
%U A070825 5963940223961341672512,8134814465483270041306368,17953535525321576981163154176,64112075360923351399733623562496,370439571435415124387660876944101888
%N A070825 One half of product of first n+1 Lucas numbers A000032.
%H A070825 Vincenzo Librandi, <a href="/A070825/b070825.txt">Table of n, a(n) for n = 0..90</a>
%H A070825 R. Grünwald, E. Heidel, A. Strätz, M. Sünkel and R. Terbach, <a href="http://www.cogsys.wiai.uni-bamberg.de/teaching/ss12/ba_pj/project_report.pdf">Induction on Number Series</a>, Fakultät fur Wirtschaftsinformatik und Angewandte Informatik, Otto-Friedrich-Universität Bamberg, 2012. - _N. J. A. Sloane_, Feb 07 2013
%F A070825 a(n) = (Product_{k=0..n} L(k))/2 with L = A000032.
%F A070825 Sum_{n>=0} 1/a(n) = 1 + A101690. - _Amiram Eldar_, Nov 09 2020
%F A070825 a(n) = 2^n*Product_{j=1..n} i^j*cosh(c*j), where c = arccsch(2) - i*Pi/2. - _Peter Luschny_, Jul 07 2025
%p A070825 c := arccsch(2) - I*Pi/2:
%p A070825 A070825 := n -> local j; 2^n*mul(I^j*cosh(c*j), j = 1..n):
%p A070825 seq(simplify(A070825(n)), n = 0..18);  # _Peter Luschny_, Jul 07 2025
%t A070825 FoldList[Times, LucasL[Range[0, 20]]]/2 (* or *)
%t A070825 Table[Round[GoldenRatio^(n(n+1)/2) QPochhammer[-1, GoldenRatio-2, n+1]]/2, {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 15 2016 *)
%o A070825 (PARI) a(n) = prod(k=0, n, fibonacci(k+1)+fibonacci(k-1))/2; \\ _Michel Marcus_, Mar 18 2016
%o A070825 (Magma) [1] cat [&*[Lucas(i+1): i in [0..n]]: n in [0..20]]; // _Vincenzo Librandi_, Sep 15 2016
%Y A070825 Cf. A000032, A003266 (for Fibonacci), A003046 (for Catalan), A101690, A135407, A218490.
%K A070825 nonn,easy
%O A070825 0,3
%A A070825 _Wolfdieter Lang_, May 10 2002